these alleged spheres, it was obvious that they could not be the
solid substance that they were commonly imagined to be, and this
fact in itself went far towards discrediting the Ptolemaic
system. It should be recalled, however, that this supposition of
tangible spheres for the various planetary and stellar orbits was
a mediaeval interpretation of Ptolemy's theory rather than an
interpretation of Ptolemy himself, there being nothing to show
that the Alexandrian astronomer regarded his cycles and epicycles
as other than theoretical.

An interesting practical discovery made by Tycho was his method
of determining the latitude of a place by means of two
observations made at an interval of twelve hours. Hitherto it had
been necessary to observe the sun's angle on the equinoctial
days, a period of six months being therefore required. Tycho
measured the angle of elevation of some star situated near the
pole, when on the meridian, and then, twelve hours later,
measured the angle of elevation of the same star when it again
came to the meridian at the opposite point of its apparent circle
about the polestar. Half the sum of these angles gives the
latitude of the place of observation.

As illustrating the accuracy of Tycho's observations, it may be
noted that he rediscovered a third inequality of the moon's
motion at its variation, he, in common with other European
astronomers, being then quite unaware that this inequality had
been observed by an Arabian astronomer. Tycho proved also that
the angle of inclination of the moon's orbit to the ecliptic is
subject to slight variation.